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As pointed out in the previous blog entry, there are, in fact, no axiomatic proofs in Greek math. But there is a widespread and sticky belief to that effect.
Why is this false belief about axiomatic proofs among Greeks so widespread and sticky? In fact, Western/church education spread the false myth.
Cambridge foolishness
Thus, on (1) that false myth of axiomatic proofs among Greeks, linked to (2) the false myth about the person Euclid and his intentions, (3) the order of theorems in the Elements was regarded as very important, and the key contribution made by “Euclid”.
This third myth was so important that the Cambridge Board of Studies foolishly laid down in its exam rules in the 1880’s that students must follow that order. This Cambridge foolishness is extraordinary because the Cambridge syndics commissioned a new text, which liberally uses empirical proofs, including, of course, the empirical proof of SAS (Side angle Side or proposition 4). Order is unimportant once an empirical proof is used: for instance the Indian proof of the “Pythagorean theorem” in the युक्तिभाषा proves the theorem in one simple step, without needing 46 earlier propositions.
The Cambridge foolishness in insisting on the order of the propositions, while using a text which gives empirical proofs tells us how the education system propagates Greediotic myths for centuries, and teaches students to ignore facts.
Church hegemony over the Western mind
Even Bertrand Russell, as a product of Cambridge, continued to believe in the “Euclid” myth of axiomatic proofs, though he realized the myth did not fit the actual book. He foolishly declared it to be Euclid’s error and not the error of the false myth of Euclid and his intentions!
That is the effect of the church control over the Western education system, and consequent hegemony over the Western mind, including the minds of those opposed to the church. This church “education” from Cambridge widely spreads myths and superstitions, which were then globalised by colonial education. It created “Greediots, Greediots everywhere and not a stop to think”.
A politically convenient reinterpretation
As Proclus explains (and the reason why he wrote his Commentary on the Elements), the Elements is a “pagan” religious text, i.e. a text on Egyptian mystery geometry which is meant to arouse the soul, exactly as Plato argued in Meno. The book Elements was never intended to be about axiomatic proofs. How did “Euclid” fit church needs to a T?
The church simply re-interpreted the book to suit its politics of reason. The church was well aware that most people are gullible, because of childhood indoctrination. And such was the fear of the church (not only the Inquisition, but even in England), that the church as well aware that no one would dare to challenge its interpretation. The facts is the no one did so for centuries.
During this time the church used the Elements to teach reasoning to its priests: a special kind of metaphysical reasoning, which suited the church, since its divorced from facts, and involving faith based or axiomatic proofs.
The church monopoly on education, through the “reputed” institutions it set up and controlled, such as Oxford and Cambridge, resulted in spreading this superstition widely among Westerners.
So widely, that when the myth of axiomatic proofs in “Euclid” ultimately collapsed (among the knowledgeable), people like Russell and Hilbert created formal mathematics to save it.

The Pythagorean calculation

Curiously, Greediots and Western historians, intent on glorifying themselves, never ever speak of the “Pythagorean CALCULATION”, though a formal proof of the “Pythagorean proposition” has no practical value, and all practical value derives from the ability to use it to CALCULATE the diagonal of a rectangle whose sides are known.
Western historians are silent about the process of calculation among Greeks. Why? (more…)

Greek history for idiots: Greediots and Pythagoras.
1: No axiomatic proofs in Greek math

Recently, I presented my talk on “Pre-colonial appropriations of Indian ganita: epistemic issues”. This was at a round table at IIAS Shimla which replaced the now-postponed conference on Indology.
The key point of my talk was that much present-day school math is an inferior sort of math which Europeans appropriated from Indian ganita without fully understanding it, and then returned during colonial times by packaging it with a false history and declaring it superior. A philosophical comparison between ganita and math was done in earlier posts and publications.
This post focuses on the false history aspect, going back to the purported Greek origins of the “Pythagorean theorem”.

False Western claim

Egyptians built massive pyramids very accurately. One would assume that to achieve that marvellous feat of engineering they knew the so-called “Pythagorean theorem”.
But in his book Mathematics in the time of the pharaohs, Richard Gillings speaks of “pyramidiots”: people who claim various sorts of wonderful knowledge is built into the pyramids of Egypt. Gillings’ argues in an appendix (citing the Greek historian Heath) that “nothing in Egyptian mathematics suggests that Egyptians were acquainted with…[even] any special case of the Pythagorean theorem.” Heath adds, “there seems to be no evidence that they [Egyptians] knew [even] that the triangle (3, 4, 5) was right-angled”. The Egyptologist Clagett chips in, “there have been exaggerated claims that Egyptians had knowledge of the Pythagorean theorem which is, of course, a formal Euclidean theorem of the Elements.”

First, Gillings, Heath etc. are not honest enough to add that there is no evidence for Pythagoras. Nor is there any evidence for the claim that he proved any sort of theorem. So, one should rightfully say, “There have been persistent false claims about a Pythagoras having proved a theorem, though there is no evidence that there was any Pythagoras nor any evidence that he proved any theorem.”
Obviously, Western history of Greeks is of very inferior quality, since the tacit norm is that stories about Greeks need no evidence and must be accepted on mere faith in Western authority: it is only stories about others which require evidence!
That is why I use the term Greediots to describe people who fantasize about all sorts of scientific achievements by Greeks without any evidence, starting from the “Pythagorean theorem”: if they can believe in that they can believe anything on their blind faith.

Religious connection of geometry

A key point: not only is there nil evidence for the story of the “Pythagorean theorem”, it is CONTRARY to all available evidence.
The Pythagoreans were a religious cult: their interest was in the connection of geometry to the RELIGIOUS belief in the soul as described by Plato, in Meno, Phaedo, Republic, etc. Anyone can check in two seconds this connection of geometry to the soul by searching for the 2^nd, 3^rd, and 4^th occurrence of “soul” in Meno, a primary source for Plato readily available online from the MIT repository. But for Greediots the story of a theorem is what is important: so they don’t and won’t check facts. (Is Plato evidence for Greek thought? If not, why has no one ever explained the grounds for rejecting Plato? And what are the other “reliable” sources, if any, for Greek history? )

Proclus, in his Commentary, explicitly asserts that this religious belief linking geometry to the soul was the sole concern of the Pythagoreans with geometry. But Greediots not only have no evidence for their beliefs, they ignore all the counter-evidence.

As Proclus further explains in his Commentary (on the book Elements today falsely attributed to an unknown “Euclid”) the book does geometry with exactly the same religious concerns. The subtle issue here is to understand Egyptian mystery geometry (and related Greek mathematics) as a sort of meditative discourse which drives the attention inwards and away from the external world.
All this is explained at great length in my book Euclid and Jesus: how and why the church changed mathematics and Christianity across two religious wars, Multiversity, 2012. See the webpage, or look inside. But Greediots will be Greediots they not only have no concern with facts they will not tolerate a counter-narrative or allow any space for it.
No axiomatic proofs in Elements
The interesting thing is how this “virgin-birth history” propagated by Greediots creates false “facts”. Clagett’s claim that “the Pythagorean theorem…is, of course, a formal Euclidean theorem of the Elements” is one such false “fact” which is widely believed.
The real fact is there is no axiomatic or formal proof of the “Pythagorean theorem” in the book Elements of “Euclid”. One has only to read the book; its very first proposition has an empirical proof not an axiomatic one. But just as most people do not read Plato, most people do not read the Elements. They just naively assume that even if the myth about its author as Euclid is false, the myth about the book must be correct. (Ha, Ha, they don’t know how thick are the layers of church lies!)
After centuries, some including Bertrand Russell finally understood the absence of axiomatic proofs in the Elements. What is shocking is for how many centuries Western scholars collectively failed to realize that even the first proposition of the Elements is contrary to the myth of axiomatic proofs in it.

(more…)

Colonial education was church education, which changed our traditional math teaching by bringing in myths and superstitions, directly related to the post-Crusade church theology of reason. Most people fail to understand this, since colonial education ensured they know nothing about (a) mathematics or its philosophy, or (b) the church theology Read more…

The NCERT class IX textbook on mathematics has its chapter 5 entitled “Euclid’s geometry”. A public grievance was lodged with the government pointing out numerous other falsehoods in the book. The grievance in the 4000 character text format specified for grievances is posted at http://ckraju.net/geometry/NCERT-grievance-note.txt. There is also a detailed Read more…